1. Field of the Invention
This invention relates to seismic-data processing, and more particularly to an improved method of migrating seismic data for steeply-dipping reflectors using an extended recursive f-k migration algorithm.
2. Description of the Related Art
Stolt (1978) method of f-k migration (frequency-wavenumber migration), incorporated herein by reference, as originally formulated, is known to be the fastest migration algorithm for 3D data volumes. However, the method requires that the acoustic velocity be constant throughout the propagation media. In order to accommodate lateral and vertical velocity variations, Stolt (1978) developed a strategy for xe2x80x9cmimickingxe2x80x9d constant velocity by time-stretching the data relative to a constant reference velocity. He also revised and then simplified the dispersion relation to reflect this stretch. The resulting equation contains an xe2x80x9cadjustment factor,xe2x80x9d W, that compensates for the stretch. The W factor defined by Stolt (1978) is a complicated function of time and space and cannot be exactly computed. In practice, a constant W based on heuristic guess is used in f-k migration. Fomel (1995, 1999), incorporated herein by reference, introduced a straight forward analytic technique for estimating W(t) from a velocity profile. However, only an average value can be used in an f-k migration, since the algorithm is performed in the frequency-wavenumber domain.
Regardless of the method by which the W is selected, f-k migration with data stretching is inaccurate for steep dips in the presence of vertical velocity variations. This is due to the fact that the method does not account for ray bending (Mikulich and Hale, 1992, incorporated herein by reference). In addition, since a constant W factor is used in the migration, the result is correct only for a very limited time range; events at earlier or later times are either over-migrated or under-migrated (Beasley et al., 1988, incorporated herein by reference). These are serious shortcomings that can be overcome to some extent in the following ways.
In one of the method to overcome foregoing shortcomings a series of constant-velocity migrations are performed using RMS velocities and the results are interpolated versus time and lateral position. One thus xe2x80x9ccarves outxe2x80x9d from the suite of 3D migrations a final, 3D volume that corresponds to the best migration at each position and time. This can also be an effective tool for pre-stack velocity estimation (Li et al., 1991, incorporated herein by reference). Unfortunately, the optimum migration velocity may differ significantly from the RMS velocity due to ray bending. The xe2x80x9ccarvingxe2x80x9d thus requires an interactive display and editing tools, since it is not viable for xe2x80x98a priorixe2x80x99 RMS velocity functions.
In another approach the ray bending effect in the f-k migration is implicitly accounted for through the use of dip-dependent velocities in the time-variable velocity function. This method was developed by Mikulich and Hale (1992), incorporated herein by reference. Unfortunately, this approach requires an inordinate computational effort.
In a yet another approach Beasley, U.S. Pat. No. 4,888,742, incorporated herein by reference, devised a scheme in which he handled a time-variable velocity function by decomposing the migration into a series of constant-velocity migrations. The migrations are thus performed in a recursive, multi-stage fashion, stripping away the portion that is completely migrated in each run. All of the sections that underlie the current stage are thus partially migrated during each migration. Velocity variation is accommodated by stretching the data relative to the constant velocity for the current stage. Ray bending is thus accommodated through the repetitive use of different residual velocities for a given time or depth interval. After each current stage has been migrated, the data are unstretched. A new stretch is then applied that is appropriate for the next migration stage. The repeated time stretching and unstretching is computationally highly burdensome.
Kim et al. (1989, 1997), incorporated herein by reference, developed another method for post-stack and pre-stack migration. In this approach Kim et al. approximate a time-variable velocity function with a coarsely-sampled, stepwise representation. The stepwise function is generated by computing a depth-time curve for vertical propagation and then approximating the curve with a set of contiguous, straight-line segments. The migration is performed in a recursive fashion, with each stage using a constant velocity. The advantages of this method are its speed and simplicity, since the data are not stretched prior to each stage. However, there is no correction for the difference between the true velocity function and its stepwise approximation. Furthermore, the method is inefficient in that it discards, for each stage, the partially migrated wavefield that lies beneath the stage that has just been fully migrated. Each stage is thus migrated using the total migration velocity for that stage. In order to prepare the wavefield for migration of the next stage, a redatuming is performed simply by a phase shift. The accommodation of ray bending is thus effected through an additional, recursive, redatuming step. This approach for handling the ray bending is computationally intensive and wasteful.
Therefore there is a continuing need for developing a method of migration that can account for ray bending and is computationally efficient.
In view of the described problems there is a continuing need for developing a method of migration of seismic event data that can account for ray bending and is computationally efficient.
A method of migrating seismic event data in the presence of a vertically time-varying velocity field defined by a migration velocity function is provided. The method comprises: approximating the migration velocity function by constant stepwise stage velocities V1ref, V2ref . . . , Vnref for stages 1 through n; time stretching the seismic event data to compensate for approximating the migration velocity function by constant stepwise stage velocities, wherein a 1st set of data comprising time-stretched seismic event data for stages 1 through n results; migrating the 1st set of data using a migration algorithm and using the migration velocity V1ref, wherein stage 1 fully migrated data results, and wherein a 2nd set of data comprising partially migrated data for stages 2 through n results; successively migrating kth through nth set of data using the migration algorithm and using a residual migration velocity that is a function of Vkref and Vkxe2x88x921ref, for k=2,3 . . . , n respectively, wherein the kth set of data comprises partially migrated data for stages k through n resulting after migrating the (kxe2x88x921)th set of data, and wherein stages k through n migrated data result; and time unstretching the stages 1 through n migrated data, wherein a seismic event migrated data results.